Evaluating a rational function at a divisor

By Luther Martin — October 8, 2009

To evaluate the pairings that we need in pairing-based cryptography, we need to evaluate the rational function that we get from a divisor at another divisor. To understand how to do this, we need to understand what it means to evaluate a rational function at a divisor. This turns out to actually be fairly straightforward.

Suppose that we have a divisor

D = Sni(Pi)

and want to evaluate a rational function f at D to get f(D). We do this by calculating

f(D) = P f(Pi)^(ni)

In the case of the divisors that we’re interested in, we’ll get a rational function of two variables x and y, and we’ll need to think of a point on an elliptic curve as being a point P = (x, y) to evaluate the rational function at the point.